## Abstract

In this paper one obtains a sequential procedure for determining the global extremum of a semi-Lipschitz real-valued function defined on a quasi-metric (asymmetric metric) space.

## Authors

**Costica Mustata**

“Tiberiu Popoviciu” Institute of Numerical Analysis Cluj-Napoca, Romanian Academy

## Keywords

Spaces with asymmetric metric; semi-Lipschitz functions; extension and approximation

### References

See the expanding block below.

## Paper coordinates

C. Mustăța, *On the approximation of the global extremum of a semi-Lipschitz function, *Mediterr. J. Math. 6 (2009), 169–180,

doi: 10.1007/s00009-009-0003-x

?

## About this paper

##### Print ISSN

1660-5446

##### Online ISSN

1660-5454

##### Google Scholar Profile

?

[1] P. Basso, Optimal search for the global maximum of functions with bounded seminorm, SIAM J. Numer. Anal. 22 (no. 5) (1985), 888–905.

[2] P. A. Borodin, The Banach-Mazur theorem for spaces with asymetric norm and its applications in convex analysis, Mat. Zametki 69 (no. 3) (2001), 329–337.

[3] S. Cobzas, Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math. 27 (no. 3) (2004), 275–296.

[4] S. Cobzas, Asymmetric locally convex spaces, Int. J. Math. Math. Sci. 16 (2005), 2585–2608.

[5] S. Cobzas and C. Mustata, Norm-preserving extyension of convex Lipschitz functions, J. Approx. Theory 24 (no. 3) (1978), 236–244.

[6] S. Cobzas and C. Mustata, Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Num´er. Th´eor. Approx. 32 (no.1) (2004), 39–50.

[7] J. Collins, and J. Zimmer, An asymmetric Arzela-Ascoli theorem, Topology Appl. 154 (no. 11) (2007), 2312–2322.

[8] P. Fletcher and W. F. Lindgren, Quasi-uniform Spaces, Marcel Dekker, New-York, 1982.

[9] S. Garcia-Ferreira, S. Romaguera and M. Sanchis, Bounded subsets and Grothendieck’s theorem for bispaces, Houston. J. Math. 25 (no. 2) (1999), 267–283.

[10] L. M. Garcia-Raffi, S. Romaguera and E. A. S´anchez-P´erez, The dual space of an asymmetric normed linear space, Quaest. Math. 26 (no. 1) (2003), 83–96.

[11] L. M. Garcia-Raffi, S. Romaguera and E. A. S´anchez-P´erez, On Hausdorff asymmetric normed linear spaces, Houston J. Math. 29 (no. 3) (2003), 717–728.

[12] M. G. Krein and A. A. Nudelman, The Markov Moment Problem and Extremum Problems, Nauka, Moscov, 1973 (in Russian), English translation: AMS, Providence, R.I., 1977.

[13] H. P. A. K¨unzi, Nonsymmetric distances and their associated topologies: about the origin of basic ideas in the area of asymmetric topology, Handbook of the History of General Topology, ed. by C.E. Aull and R. Lower, vol. 3, Hist. Topol. 3, Kluwer Acad. Publ. (Dordrecht, 2001), 853–968.

[14] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837–842.

[15] A. Mennucci, On asymmetric distances, Technical report, Scuola Normale Superiore, Pisa, 2004.

[16] C. Mustata, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory 19 (no. 3) (1977), 222–230.

[17] C. Mustata, Extension of H¨older Functions and some related problems of best approximation, ”Babes-Bolyai” University, Faculty of Mathematics, Research Seminars, Seminar onf Mathematical Analysis (1991) 71–86.

[18] C. Mustata, Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx. 30 (no. 1) (2001), 61–67.

[19] C. Mustata, On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx. 31 (no. 1) (2002), 61–67.

[20] V. Pestov and A. Stojmirovic, Indexing schemes for similarity search: an illustrated paradigm, Fund. Inf. 70 (no. 4) (2006), 367–385.

[21] S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory 103 (2000), 292–301.

[22] S. Romaguera and M. Sanchis, Properties of the normed cone of semi-Lipschitz functions, Acta Math. Hungar 108 (nos. 1-2) (2005), 55–70.

[23] S. Romaguera, J. M. Sanchez-Alvarez and M. Sanchis, ´ El espacio de funciones semiLipschitz, VI Jornadas de Matem´atica Aplicada, Departamento de Matem´atica Aplicada, Universidad Polit´ecnica de Valencia, 1-3 septiembrie, 2005.

[24] J. M. Sanchez-Alvarez, On semi-Lipschitz functions with values in a quasi-normed linear space, Appl. Gen. Top. 6 (no. 2) (2005), 217–228.

[25] B. Shubert, A sequential method seeking the global maximum of a function, SIAM J. Num. Anal. 9 (1972), 379–388.

[26] A. Stojmirovic, Quasi-metric spaces with measures, Proc. 18th Summer Conference on Topology and its Applications, Topology Proc. 28 (no. 2) (2004), 655–671.

[27] W. A. Wilson, On quasi-metric spaces, Amer. J. Math. 53 (no. 3) (1931), 75–684.